Subgroups — #2091×C₁₃₇ #31777102

Order 17536.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430×C₁₃₇	C₂	1	3	yes	maximal, complemented
2	#407×C₁₃₇	C₂	1	3	yes	maximal
3	#338×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₁₃₇	C₂²	1	3	yes	—
6	C₂×M₁₆×C₁₃₇	C₂²	1	3	yes	—
7	#108×C₁₃₇	—	2	6	no	—
8	C₂²⋊C₈×C₁₃₇	—	2	6	no	—
9	C₂⁴×C₁₃₇	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₁₃₇	—	2	3	no	—
11	C₂²×C₄×C₁₃₇	—	2	3	no	—
12	C₂²×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
13	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
14	M₁₆×C₁₃₇	—	2	6	no	—
15	M₁₆×C₁₃₇	—	2	6	no	—
16	C₂×C₈×C₁₃₇	—	2	3	no	—
17	C₂×C₈×C₁₃₇	D₄	1	6	yes	—
18	C₂³×C₁₃₇	—	2	3	no	—
19	C₂³×C₁₃₇	—	4	2	no	—
20	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	—	2	3	no	—
22	C₂×C₄×C₁₃₇	—	4	3	no	—
23	C₂×C₄×C₁₃₇	—	4	1	no	—
24	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₁₃₇	#41	1	3	yes	—
26	C₈×C₁₃₇	—	2	6	no	—
27	C₈×C₁₃₇	—	4	3	no	—
28	C₂²×C₁₃₇	—	2	6	no	—
29	C₂²×C₁₃₇	—	4	1	no	—
30	C₂²×C₁₃₇	—	4	3	no	—
31	C₂²×C₁₃₇	—	4	1	no	—
32	C₂²×C₁₃₇	C₂²⋊Q₈	1	3	yes	—
33	C₄×C₁₃₇	—	4	1	no	—
34	C₄×C₁₃₇	#123	1	3	yes	—
35	C₄×C₁₃₇	#127	1	1	yes	characteristic
36	C₂×C₁₃₇	—	2	3	no	—
37	C₂×C₁₃₇	—	4	2	no	—
38	C₂×C₁₃₇	#399	1	1	yes	characteristic
39	C₁₃₇	#2091	1	1	yes	characteristic, direct product factor, Sylow
40	#2091	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	#430	C₂×C₁₃₇	1	3	yes	complemented
42	#407	C₂×C₁₃₇	1	3	yes	—
43	#338	C₂×C₁₃₇	1	1	yes	characteristic, complemented
44	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
45	C₂×M₁₆	C₂²×C₁₃₇	1	3	yes	—
46	C₂×M₁₆	C₂²×C₁₃₇	1	3	yes	—
47	#108	—	2	6	no	—
48	C₂²⋊C₈	—	2	6	no	—
49	C₂⁴	Q₈×C₁₃₇	1	1	yes	characteristic
50	C₂²×C₄	—	2	3	no	—
51	C₂²×C₄	—	2	3	no	—
52	C₂²×C₄	Q₈×C₁₃₇	1	1	yes	characteristic
53	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	2	6	no	—
56	C₂×C₈	—	2	3	no	—
57	C₂×C₈	D₄×C₁₃₇	1	6	yes	—
58	C₂³	—	2	3	no	—
59	C₂³	—	4	2	no	—
60	C₂³	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
61	C₂×C₄	—	2	3	no	—
62	C₂×C₄	—	4	3	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
65	C₂×C₄	#41×C₁₃₇	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	4	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	3	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊Q₈×C₁₃₇	1	3	yes	—
73	C₄	—	4	1	no	—
74	C₄	#123×C₁₃₇	1	3	yes	—
75	C₄	#127×C₁₃₇	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	4	2	no	—
78	C₂	#399×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₇
socle	C₂×C₁₃₇	#399
prime radical	C₂⁴×C₁₃₇	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #430×C₁₃₇	C₂	semidirect
3. #338×C₁₃₇	C₂	semidirect
39. C₁₃₇	#2091	direct (N × H)
40. #2091	C₁₃₇	direct (N × H)
41. #430	C₂×C₁₃₇	semidirect
43. #338	C₂×C₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#430×C₁₃₇	2
#407×C₁₃₇	2
#338×C₁₃₇	2
#2091	137

Subgroups — #2091×C137 #31777102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2091×C₁₃₇ #31777102